New versions of Barbalat’s lemma with applications

Hou, Ming; Duan, Guang‐Ren; Guo, Mengshu

doi:10.1007/s11768-010-8178-z

Cited by 64 publications

(29 citation statements)

References 4 publications

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“…It can obviously conclude that {z 1 , z 2 , z 3 , z 4 , θ 1 , θ 2 } ∈ 0 .It is also easy to check that V is bounded. So, V → 0 as t → ∞ by the Barbalat's lemma[32].…”

mentioning

confidence: 86%

Adaptive decoupling synchronous control of dissimilar redundant actuation system for large civil aircraft

Shi

Wang

et al. 2015

Aerospace Science and Technology

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“…It can obviously conclude that {z 1 , z 2 , z 3 , z 4 , θ 1 , θ 2 } ∈ 0 .It is also easy to check that V is bounded. So, V → 0 as t → ∞ by the Barbalat's lemma[32].…”

mentioning

confidence: 86%

Adaptive decoupling synchronous control of dissimilar redundant actuation system for large civil aircraft

Shi

Wang

et al. 2015

Aerospace Science and Technology

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“…Proof From the above Lyapunov function V = Y 1 ( L ), the time derivative of V is available

\begin{matrix} right \dot{V} = & left \frac{2 L}{(B - A) (δ - L^{2})} [2 ė - \dot{A} - \dot{B} - L (\dot{B} - \dot{A})] & right \\ right = & left \frac{2 L}{(B - A) (δ - L^{2})} [2 u - 2 \dot{r} - \dot{A} - \dot{B} - L (\dot{B} - \dot{A})] . \end{matrix}

Due to

u = - \frac{1}{4} L false(B - A false) + \overset{r}{true} + \frac{1}{2} false[\overset{A}{true} + \overset{B}{true} + L false(\overset{B}{true} - \overset{A}{true} false) false]

and according to Lemma ,

\overset{V}{true} = - \frac{L^{2}}{δ - L^{2}} \leq - \ln \frac{δ}{δ - L^{2}} = - V .

From Equation , we can get the following two conclusions. (1) From and Barbalat's Lemma in the work of Hou et al, we know

\begin{matrix} right \lim_{t \to \infty} \ln δ \end{matrix}

…”

Section: Prescribed Performance Controlmentioning

confidence: 94%

Robust adaptive prescribed performance control for a class of nonlinear pure‐feedback systems

Jia

Wang

Zhou

2019

Intl J Robust & Nonlinear

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Summary An adaptive prescribed performance control design procedure for a class of nonlinear pure‐feedback systems with both unknown vector parameters and unmodeled dynamics is presented. The unmodeled dynamics lie within some bounded functions, which are assumed to be partially known. A state transformation and an auxiliary system are proposed to avoid using the cumbersome formula to handle the nonaffine structure. Simultaneously, a parameter‐type Lyapunov function and L function are designed to ensure the prescribed performance of the pure‐feedback system. As illustrated by examples, the proposed adaptive prescribed performance control scheme is shown to guarantee global uniform ultimate boundedness. At the same time, this method not only guarantees the prescribed performance of the system but also makes the tracking error asymptotically close to a certain value or stable.

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“…Since is bounded, we get that ( , ) is bounded, then ( , ) is uniformly continuous. from the Barbalat Lemma [24], we can get:…”

Section: Design Of the Finite Time Control Lawmentioning

confidence: 99%

Finite time point-stabilization of underwater spherical roving robot

Liu¹,

Li²,

Jia³

et al. 2016

J. vibroeng.

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This paper addresses the point stabilization problem for the underwater spherical roving robot (BYSQ-3) in the horizontal plane. The finite-time stable control laws are adopted to steer the robot to the origin fast, accurately and reliably. Firstly, the inner structure and operational principle of the robot is described and the kinematic and dynamic equations are established. Secondly, the diffeomorphism transformation and change of inputs are introduced to decouple the multivariable coupling system into two subsystems. The second subsystem consists of two double integrator systems. The finite-time controller is introduced to ensure part states converge to zero in finite time. Then, the other states are steered to the origin using the same method. Thirdly, the design process has no virtual input and the stability analysis is simple, the controller designed is easy for engineering implementation. The simulation and experiment results are presented to validate the shorter convergence time and better stability character of the controller.

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New versions of Barbalat’s lemma with applications

Abstract: This note presents a set of new versions of Barbalat's lemma combining with positive (negative) definite functions. Based on these results, a set of new formulations of Lyapunov-like lemma are established. A simple example shows the usefulness of our results.

Cited by 64 publications

References 4 publications

Adaptive decoupling synchronous control of dissimilar redundant actuation system for large civil aircraft

Adaptive decoupling synchronous control of dissimilar redundant actuation system for large civil aircraft

Robust adaptive prescribed performance control for a class of nonlinear pure‐feedback systems

Finite time point-stabilization of underwater spherical roving robot

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